Types of numbers at the time of Liouville: Integers, Rational Numbers, Algebraic numbers (roots of polynomials with rational coefficients), transcendental numbers (numbers that are not roots of any polynomial with rational coefficients)

The concept of transcendental numbers existed before any were known

Liouville proved that you can't approximate any algebraic number really well with rational numbers

Theorem: If you find a sequence $\frac{p_m}{q_m} \to \alpha$ such that $\mid \alpha - \frac{p_m}{q_m} \mid \leq \frac{1}{{q_m}^n}$ for $m$ sufficiently large then $\alpha$ is not the root of a polynomial of degree $n$